Dynamic and Static Nature of Br4σ(4c–6e) and Se2Br5σ(7c–10e) in the Selenanthrene System and Related Species Elucidated by QTAIM Dual Functional Analysis with QC Calculations

Hayashi, Satoko; Nishide, Taro; Nakanishi, Waro

doi:https://doi.org/10.1155/2020/2901439

Bioinorganic Chemistry and Applications

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Research Article | Open Access

Volume 2020 | Article ID 2901439 | https://doi.org/10.1155/2020/2901439

Dynamic and Static Nature of Br₄σ(4c–6e) and Se₂Br₅σ(7c–10e) in the Selenanthrene System and Related Species Elucidated by QTAIM Dual Functional Analysis with QC Calculations

Satoko Hayashi,¹Taro Nishide,¹and Waro Nakanishi¹

Academic Editor: Claudio Pettinari

Received22 Jan 2019

Revised15 Jun 2019

Accepted19 Nov 2019

Published24 Jul 2020

Abstract

The nature of Br₄σ(4c–6e) of the ^BBr-∗-^ABr-∗-^ABr-∗-^BBr form is elucidated for SeC₁₂H₈(Br)SeBr---Br-Br---BrSe(Br)C₁₂H₈Se, the selenanthrene system, and the models with QTAIM dual functional analysis (QTAIM-DFA). Asterisks (∗) are employed to emphasize the existence of bond critical points on the interactions in question. Data from the fully optimized structure correspond to the static nature of interactions. In our treatment, data from the perturbed structures, around the fully optimized structure, are employed for the analysis, in addition to those from the fully optimized one, which represent the dynamic nature of interactions. The ^ABr-∗-^ABr and ^ABr-∗-^BBr interactions are predicted to have the CT-TBP (trigonal bipyramidal adduct formation through charge transfer) nature and the typical hydrogen bond nature, respectively. The nature of Se₂Br₅σ(7c–10e) is also clarified typically, employing an anionic model of [Br-Se(C₄H₄Se)-Br---Br---Br-Se(C₄H₄Se)-Br]⁻, the 1,4-diselenin system, rather than (BrSeC₁₂H₈)Br---Se---Br-Br---Br-Se(C₁₂H₈Se)-Br, the selenanthrene system.

1. Introduction

We have been much interested in the behavior of the linear interactions of the σ-type, higher than σ(3c–4e: three center-four electron interactions) [1–6], constructed by the atoms of heavier main group elements. We proposed to call such linear interactions the extended hypervalent interactions, σ(mc–ne: 4 ≤ m; m < n < 2m), after the hypervalent σ(3c–4e). The linear alignments of four chalcogen atoms were first demonstrated in the naphthalene system, bis[8-(phenylchalcogenyl)naphthyl]-1,1′-dichalcogenides [I: 1-(8-Ph^BEC₁₀H₆)^AE-^AE(C₁₀H₆^BEPh-8′)-1′ (^AE, ^BE = S and Se)] [7–12]. It was achieved through the preparation and the structural determination by the X-ray crystallographic analysis. The linear ^BE---^AE-^AE---^BE interactions in I are proposed to be analysed as the σ(4c–6e) model not by the double ^AE^BE₂σ(3c–4e) model. σ(4c–6e) in I is characterized by the CT interaction of the n_p(^BE) ⟶ σ∗(^AE–^AE)←n_p(^BE) form [8, 10–12], where n_p(^BE) stands for the p-type nonbonding orbitals of ^BE and σ∗(^AE-^AE) are the σ∗ orbitals of ^AE-^AE. The novel reactivity of σ(4c–6e) in I was also clarified [8].

σ(4c–6e) is the first member of σ(mc–ne: 4 ≤ m; m < n < 2m) [7–13]. The σ(4c–6e) interactions are strongly suggested to play an important role in the development of high functionalities in materials and in the key processes of biological and pharmaceutical activities, recently. The bonding is applied to a wide variety of fields, such as crystal engineering, supramolecular soft matters, and nanosciences [4, 14–23]. The nature of ^BE---^AE and ^AE-^AE in ^BE---^AE-^AE---^BE of σ(4c–6e) has been elucidated [24–27] using the quantum theory of atoms in molecules (QTAIM) approach, introduced by Bader [28–37]. The linear interactions of the σ(4c–6e) type will form if ^BE in is replaced by X, giving E₂X₂σ(4c–6e). The nature of E₂X₂σ(4c–6e) in the naphthalene system of 1-(8-XC₁₀H₆)E-E(C₁₀H₆X-8′)-1′ [II (E, X) = (S, Cl), (S, Br), (Se, Cl), and (Se, Br)] was similarly clarified very recently [38].

The σ(4c–6e) interaction will also be produced even if both ^BE and ^AE in are replaced by X. X₄σ(4c–6e) should also be stabilized through CT of the n_p(X) ⟶ σ∗(X-X) ← n_p(X) form. The energy lowering of the system through the CT interaction must be the driving force for the formation of X₄σ(4c–6e). X₄σ(4c–6e) is the typical kind of halogen bonds, together with E₂X₂σ(4c–6e), which are of current and continuous interest [39]. Br₄σ(4c–6e) has been clearly established in the selenanthrene system, SeC₁₂H₈(Br)SeBr---Br-Br---BrSe(Br)C₁₂H₈Se (1), through the preparation and the structural determination by the X-ray crystallographic analysis [39]. The atoms taking part in the linear interaction in question are shown in bold. The structure of (BrSeC₁₂H₈)Br---Se---Br-Br---Br-Se(C₁₂H₈Se)-Br (2) was also reported, in addition to 1, which is suggested to contain Se₂Br₅σ(7c–10e) since the seven atoms of Se₂Br₅ align almost linearly in crystals. Figure 1 shows the structures of 1 and 2 determined by the X-ray analysis and the approximate MO model for σ(4c–6e) and σ(7c–10e).

(a)

(b)

(c)

(d)

It is challenging to elucidate the nature of Br₄σ(4c–6e) of the n_p(Br) ⟶ σ∗(Br-Br)←n_p(Br) form in 1 and Se₂Br₅σ(7c–10e) in 2, together with the related species. Figure 2 illustrates the process assumed for the formation of 1 and 2 from selenanthrene (S: SeC₁₂H₈Se). In this process, (SeC₁₂H₈)Br-Se-Br (3) should be formed first in the reaction of S with Br₂, and then 3 reacts with Br₂ to yield Br[Se(Br) (C₁₂H₈)]Se---Br-Br (4). The almost linear alignment of Br---Se---Br-Br in 4 could be analysed by the SeBr₃σ(4c–6e) model, where the Br and Se atoms in 4 are placed in close proximity in space. While 1 containing Br₄σ(4c–6e) forms in the reaction of (3 + Br₂ + 3), the reaction of 3 + 4 yields 2, consisting Se₂Br₅σ(7c–10e). Both 1 and 2 are recognized as the Br₂-included species. While XC₄H₄(Br)SeBr---Br-Br---BrSe(Br)C₄H₄X (5 (X = Se) and 6 (X = S)), models of 1, also consisted of Br₄σ(4c–6e), Se₂Br₅σ(7c–10e) will appear typically in the anionic species, [Br-Se(Me₂)-Br---Br---Br-Se(Me₂)-Br]⁻ (7) and [Br-Se(SeC₄H₄)-Br---Br---Br-Se(C₄H₄Se)-Br]⁻ (8), models of 2. Species, 5, 6, 7, and 8, are shown in Figure 2, where 5, 6, and 8 belong to the 1,4-diselenin system.

What are the differences and similarities between X₄σ(4c–6e), E₄σ(4c–6e), and E₂X₂σ(4c–6e)? The nature of X₄σ(4c–6e) in 1 (X = Br) is to be elucidated together with the models. Models, other than 5 and 6, are also devised to examine the stabilization sequence of Br₄σ(4c–6e). H₂Br₄ (C_2h) and Me₂Br₄ (C_2h) have the form of R-Br---Br-Br---Br-R (RBr₄R: R = H and Me), which are called the model group A (G(A)). The electronic efficiency to stabilize Br₄σ(4c–6e) seems small for R in G(A). Br₆ (C_2h) is detected as the partial structure in the crystals of Br₂ [40]. Br₆ (C_2h) in the crystals is denoted by Br₆ (C_2h)_obsd. The optimized structure of Br₆ (C_2h) has one imaginary frequency, which belongs to G(A), together with Br₆ (C_2h)_obsd. The optimized structure of Br₆ retains the C₂ symmetry, (Br₆ (C₂)), which also belongs to G(A). The CT interaction of the n_p(^BBr) ⟶ σ∗(^ABr-^ABr) ⟵ n_p(^BBr) form in Br₄σ(4c–6e) will be much stabilized if the large negative charge is developed at the ^BBr atoms in Br-(R₂)Se-^BBr---^ABr-^ABr---^BBr-Se(R₂)-Br, where the ∠Se^BBr^ABr is around 90°. The highly negatively charged ^BBr in Br-Se(R₂)-^BBr (R = H and Me) of σ(3c–4e) is employed to stabilize Br₄σ(4c–6e), in this case. The models form G(B). The nature of Br₄σ(4c–6e) in 5 and 6 is similarly analysed, which belongs to G(B). (D_∞h) also belongs to G(B) although one imaginary frequency was predicted for , if optimized at the MP2 level. Figure 3 illustrates the story for the stabilization of Br₄σ(4c–6e) in the sequence of the species, starting from G(A) to 1, via G(B). Figure 3 also shows the ^ABr-^ABr and ^ABr---^BBr distances (r(^ABr-^ABr) and r(^ABr-^BBr), respectively), together with the charge developed at ^BBr in the original species of R-^BBr (Qn (^BBr)), which construct R-^BBr---^ABr-^ABr---^BBr-R.

A chemical bond or interaction between atoms A and B is denoted by A-B, which corresponds to a bond path (BP) in the quantum theory of atoms in molecules (QTAIM) approach, introduced by Bader [28–37]. We will use A-∗-B for BP, where the asterisk emphasizes the existence of a bond critical point (BCP, ∗) in A-B [28, 29]. (Dots are usually employed to show BCPs in molecular graphs. Therefore, A-•-B would be more suitable to describe the BP with a BCP. Nevertheless, A-∗-B is employed to emphasize the existence of a BCP on the BP in question in our case. BCP is a point along BP at the interatomic surface, where ρ(r) (charge density) reaches a minimum along the interatomic (bond) path, while it is a maximum on the interatomic surface separating the atomic basins). The chemical bonds and interactions are usually classified by the signs of Laplacian rho (∇²ρ_b(r_c)) and H_b(r_c) at BCPs, where ρ_b(r_c) and H_b(r_c) are the charge densities and total electron energy densities at BCPs, respectively (see Scheme S1 in Supplementary File). The relations between H_b(r_c), ∇²ρ_b(r_c), G_b(r_c) (the kinetic energy densities), and V_b(r_c) (the potential energy densities) are represented in equations (1) and (2):

How can the nature of Br₄σ(4c–6e) and Se₂Br₅σ(7c–10e) be clarified? For the characterization of interactions in more detail, we recently proposed QTAIM dual functional analysis (QTAIM-DFA) [42–47] for experimental chemists to analyze their own chemical bonds and interaction results based on their own expectations, according to the QTAIM approach [28–37]. H_b(r_c) is plotted versus H_b(r_c) − V_b(r_c)/2 (= (ћ²/8m)∇²ρ_b(r_c)) at BCPs in QTAIM-DFA. The classification of interactions by the signs of ∇²ρ_b(r_c) and H_b(r_c) is incorporated in QTAIM-DFA. Data from the fully optimized structures correspond to the static natures of the interactions, which are analysed using the polar coordinate (R, θ), representation [42, 44–46]. Each interaction plot, containing data from both the perturbed structures and the fully optimized one include a specific curve that provides important information about the interaction. This plot is expressed by (θ_p, κ_p), where θ_p corresponds to the tangent line of the plot and κ_p is the curvature. The concept of the dynamic nature of interactions has been proposed based on (θ_p, κ_p) [42, 44]. θ and θ_p are measured from the y-axis and the y-direction, respectively. We call (R, θ) and (θ_p, κ_p) QTAIM-DFA parameters, which are drawn in Figure 4, exemplified by (D_∞h). While (R, θ) classifies the interactions, (θ_p, κ_p) characterizes them.

(a)

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We proposed a highly reliable method to generate the perturbed structures for QTAIM-DFA very recently [48]. The method is called CIV, which employs the coordinates derived from the compliance force constants C_ij for the internal vibrations. Compliance force constants C_ij are defined as the partial second derivatives of the potential energy due to an external force, as shown in equation (3), where i and j refer to the internal coordinates and the force constants f_i and f_j correspond to i and j, respectively. The C_ij values and the coordinates corresponding to the values can be calculated using the compliance 3.0.2 program, released by Brandhorst and Grunenberg [49–52]. The dynamic nature of interactions based on the perturbed structures with CIV is described as the “intrinsic dynamic nature of interactions” since the coordinates are invariant to the choice of the coordinate system:

QTAIM-DFA has excellent potential for evaluating, classifying, characterizing, and understanding weak to strong interactions according to a unified form. The superiority of QTAIM-DFA to elucidate the nature of interactions, employing the perturbed structures generated with CIV, is explained in the previous papers [48, 53] (see also Figure S2 and Table S2 in Supplementary File). QTAIM-DFA is applied to standard interactions and rough criteria that distinguish the interaction in question from others which are obtained. QTAIM-DFA and the criteria are explained in Supplementary File using Schemes S1–S3, Figures S1 and S2, Table S1, and equations (S1)–(S7). The basic concept of the QTAIM approach is also explained.

We consider QTAIM-DFA, employing the perturbed structures generated with CIV, to be well suited to elucidate the nature of Br₄σ(4c–6e) in 1, Se₂Br₅σ(7c–10e) in 2, and the models derived from 1 and 2, together with the related linear interactions. The interactions in Br₄σ(4c–6e) are denoted by ^BBr-∗-^ABr-∗-^ABr-∗-^BBr, where the asterisk emphasizes the existence of a BCP in the interactions, so are those in Se₂Br₅σ(7c–10e). Herein, we present the results of the investigations on the extended hypervalent interactions in the species, together with the structural feature. Each interaction is classified and characterized, employing the criteria as a reference.

2. Methodological Details in Calculations

Calculations were performed employing the Gaussian 09 programs package [54]. The basis sets employed for the calculations were obtained, as implemented from Sapporo Basis Set Factory [55]. The basis sets of the (621/31/2), (6321/621/3), (74321/7421/72), and (743211/74111/721/2+1s1p) forms were employed for C, S, Se, and Br, respectively, with the (31/3) form for H. The basis set system is called BSS-A. All species were calculated employing BSS-A, and the Møller–Plesset second-order energy correlation (MP2) level [56–58] was applied for the optimizations. Optimized structures were confirmed by the frequency analysis. The results of the frequency analysis were used to calculate the C_ij values and the coordinates (C_i) corresponding to the values. The DFT level of CAM-B3LYP [59] was also applied when necessary. The QTAIM functions were analysed with the AIM2000 [60] and AIMAll [61] programs.

The method to generate perturbed structures with CIV is the same as that explained in the previous papers [48, 53]. As shown in equation (4), the i-th perturbed structure in question (S_iw) is generated by the addition of the i-th coordinates (C_i), derived from C_ij, to the standard orientation of a fully optimized structure (S_o) in the matrix representation. The coefficient f_iw in equation (4) controls the structural difference between S_iw and S_o: f_iw is determined to satisfy equation (5) for r, where r and r_o stand for the interaction distances in question in the perturbed and fully optimized structures, respectively, with a_o = 0.52918 Å (Bohr radius). The C_i values of five digits are used to predict S_iw:

In QTAIM-DFA, H_b(r_c) is plotted versus H_b(r_c) − V_b(r_c)/2 for data of = 0, ±0.05, and ±0.10 in equation (5). Each plot is analysed using a regression curve of the cubic function, as shown in equation (6), where (x, y) = (H_b(r_c) − V_b(r_c)/2 and H_b(r_c)) ( (square of correlation coefficient) > 0.99999 in usual) [46].

3. Results and Discussion

3.1. Structural Optimizations

The structures of 1 (C_i) and 2 (C₁) determined by the X-ray analysis are denoted by 1 (C_i)_obsd and 2 (C₁)_obsd, respectively [39]. The structural parameters are shown in Tables S2 and S3 in Supplementary File, respectively. Figure 3 contains the selected structural parameters for 1 (C_i)_obsd. The structures are optimized for G(A) of H₂Br₄ (C_2h), Me₂Br₄ (C_2h), Br₆ (C_2h), and Br₆ (C₂) and G(B) of H₄Se₂Br₆ (C_i), Me₄Se₂Br₆ (C_i), 5 (C_i), and 6 (C_i), together with 3 (C_s), 4 (C_s), 7 (C_2h), 8 (C_2h), and Br₂ (D_∞h). The optimized structural parameters are also collected in Tables S2 and S3 in Supplementary File. The frequency analysis was successful for the optimized structures, except for 1 (C_i)_obsd and Br₆ (C_2h). All positive frequencies were obtained for 1 (C_i), if calculated with CAM-B3LYP/BSS-A, which confirms the structure. The Br---Br distances of Br₄σ(4c–6e) in 1 (C_i) are somewhat longer if optimized at the CAM-B3LYP level, relative to 1 (C_i)_obsd. While one imaginary frequency is detected in Br₆ (C_2h), Br₆ (C₂) has all positive frequencies. The optimized structures are not shown in figures, instead, some of them can be found in Figures 3 and 5, where the molecular graphs are drawn on the optimized structures. Figure 3 contains the optimized r(^ABr-^ABr) and r(^ABr-^BBr) distances for the models and the charge developed at ^BBr in the original R-^BBr and Br-(R₂)Se-^BBr (Qn (^BBr)), which give the models of G(A) and G(B), respectively. The r(^ABr-^BBr) values become shorter in the order shown in equation (7), if evaluated with MP2/BSS-A:

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Figure 5

Molecular graphs of 5 (C_i) (a), 6 (C_i) (b), 7 (C_2h) (c), and 8 (C_2h) (d) drawn on the structures optimized at the MP2 level, together with 1 (C_i)_obsd (e) and 2 (C₁)_obsd (f). Contour plots of ρ(r) are also drawn on the planes containing the linear interactions. BCPs are denoted by red dots, RCPs (ring critical points) by yellow dots, and CCPs (cage critical points) by green dots. BPs (bond paths) are drawn as pink lines and the secondary ones as pink dots. They are associated with the BCPs. Carbon and hydrogen atoms are shown in black and gray, respectively. The contours () are at 2^l (l = ±8, ±7, …, and 0).

One imaginary frequency was also predicted for (D_∞h) if optimized with MP2/BSS-A. (D_∞h) seems to collapse to and Br⁻, according to the imaginary frequency. The double negative charges in (D_∞h) would be responsible for the results. The electrostatic repulsion between the double negative charges will operate to collapse it.

3.2. Energies for Formation of Br₄σ(4c–6e) and NBO Analysis

Energies for the formation of R′Br₄R′ from the components (2R′Br + Br₂) (ΔE) are defined by equation (8). The ΔE values evaluated on the energy surface are denoted by ΔE_ES, while those corrected with the zero-point energies are by ΔE_ZP. The ΔE_ES and ΔE_ZP values for the optimized structures are given in Table S2 in Supplementary File. ΔE_ZP are excellently correlated to ΔE_ES (ΔE_ZP = 0.99ΔE_ES + 1.93: R_c² = 0.9998, see Figure S3 in Supplementary File):

NBO analysis [62] was applied to ^ABr---^BBr of the species to evaluate the contributions from CT to stabilize R′-^BBr---^ABr-^ABr---^BBr-R′. For each donor NBO (i) and acceptor NBO (j), the stabilization energy E(2) is calculated based on the second-order perturbation theory in NBO, according to equation (9), where q_i is the donor orbital occupancy, ε_i and ε_j are diagonal elements (orbital energies), and F(i, j) is the off-diagonal NBO Fock matrix element. The results are collected in Table S4 in Supplementary File. The ΔE_ES values are very well correlated to E(2) for the optimized structures, except for (D_∞h). (ΔE_ES = –0.71(2E(2)) + 7.17: = 0.959, see Figure S4 in Supplementary File). (D_∞h) is predicted to be less stable than the components.

Before application of QTAIM-DFA to Br₄σ(4c–6e) and Se₂Br₅σ(7c–10e), molecular graphs were examined, as shown in the next section.

3.3. Molecular Graphs with Contour Plots for the Species Containing Br₄σ(4c–6e), Se₂Br₅σ(7c–10e), and Related Linear Interactions

Figure 5 illustrates the molecular graphs of 5 (C_i), 6 (C_i), 7 (C_2h), and 8 (C_2h), drawn on the optimized structures, together with 1 (C_i)_obsd and 2 (C₁)_obsd. Figure 5 also shows the contour plots of ρ(r) drawn on the suitable plane in the molecular graphs. BCPs are well demonstrated to locate on the (three-dimensional) saddle points of ρ(r). Molecular graphs of Me₂Br₄ (C_2h), Br₆ (C₂), (D_∞h), and Br(Me₂)SeBr₄Se(Me₂)Br (C_i) are shown in Figure 3, which are drawn on the optimized structures.

3.4. Survey of Br₄σ(4c–6e) and Se₂Br₅σ(7c–10e)

BPs in Br₄σ(4c–6e) and Se₂Br₆σ(7c–10e) seem straight, as shown in Figures 3 and 5. To show the linearity more clearly, the lengths of BPs (r_BP) for Br₄σ(4c–6e) are calculated. The values are collected in Table S5 in Supplementary File, together with the corresponding straight-line distances (R_SL). The table contains the values for Se₂Br₆σ(7c–10e) in 7 (C_2h) and 8 (C_2h). The differences between them (Δr_BP = r_BP–R_SL) are less than 0.003 Å. The r_BP values are plotted versus R_SL, which are shown in Figure S5 in Supplementary File. The correlations are excellent, as shown in the figure. Therefore, Br₄σ(4c–6e) and Se₂Br₆σ(7c–10e) in the species can be approximated by the straight lines.

QTAIM functions are calculated for Br₄σ(4c–6e) at BCPs. Table 1 collects the values for the interactions. H_b(r_c) is plotted versus H_b(r_c) − V_b(r_c)/2 for the data shown in Table 1, together with those from the perturbed structures generated with CIV. Figure 4 shows the plots for the ^ABr-∗-^ABr and ^ABr-∗-^BBr interactions in Br₄σ(4c–6e) of the bromine species. The plots for ^ABr-∗-^ABr appear in the region of H_b(r_c) − V_b(r_c)/2 > 0 and H_b(r_c) < 0, for all species, except for the original Br₂ (D_∞h), of which the plot appears in the region of H_b(r_c) − V_b(r_c)/2 < 0 and H_b(r_c) < 0. Therefore, the interactions are all classified by the regular-CS (closed shell) interactions, except for Br₂ (D_∞h), which is classified by the SS (shard shell) interaction. On the contrary, data of ^ABr-∗-^BBr appear in the region of H_b(r_c) − V_b(r_c)/2 > 0 and H_b(r_c) > 0 for all species, except for those in H₄Se₂Br₆ (C_i), Me₄Se₂Br₆ (C_i), 5 (C_i), and 6 (C_i), which appear in the region of H_b(r_c) − V_b(r_c)/2 > 0 and H_b(r_c) < 0. As a result, ^ABr-∗-^BBr is classified by the pure-CS interactions (p-CS) for all, except for the four species, of which ^ABr-∗-^BBr is classified by the regular-CS interactions (r-CS). The ^ABr-∗-^BBr interaction in (D_∞h) is very close to the borderline between p-CS and r-CS since H_b(r_c) = 0.0001 au for (D_∞h), which is very close to zero. QTAIM-DFA parameters of (R, θ) and (θ_p, κ_p) are obtained by analysing the plots of H_b(r_c) versus H_b(r_c) − V_b(r_c)/2 in Figure 4, according to equations (S3)–(S6). Table 1 collects the QTAIM-DFA parameters for Br₄σ(4c–6e). The classification of interactions will also be discussed based on the (R, θ) values.

QTAIM functions are similarly calculated for Se₂Br₆σ(7c–10e) at BCPs, together with the related interactions. H_b(r_c) is similarly plotted versus H_b(r_c) − V_b(r_c)/2 although not shown in the figures. Then, QTAIM-DFA parameters of (R, θ) and (θ_p, κ_p) are obtained by analysing the plots, according to equations (S3)–(S6). Table 2 collects the QTAIM-DFA parameters of (R, θ) and (θ_p, κ_p) for Br₄σ(4c–6e).

3.5. Nature of Br₄σ(4c–6e)

Interactions are characterized by (R, θ), which correspond to the data from the fully optimized structures. On the contrary, they are characterized employing (θ_p, κ_p) derived from the data of the perturbed structures around the fully optimized structures and the fully optimized ones. In this case, the nature of interactions is substantially determined based of the (R, θ, θ_p) values, while the κ_p values are used only additionally. It is instructive to survey the criteria before detail discussion. The criteria tell us that 180° < θ (H_b(r_c) − V_b(r_c)/2 < 0) for the SS interactions, 90° < θ < 180° (H_b(r_c) < 0) for the r-CS interactions, and 45° < θ < 90° (H_b(r_c) > 0) for p-CS interactions. The θ_p value characterizes the interactions. In the p-CS region of 45° < θ < 90°, the character of interactions will be the vdW type for 45° < θ_p < 90°, whereas it will be the typical HB type without covalency (t-HB_nc) for 90° < θ_p < 125°, where θ_p = 125° is tentatively given for θ = 90°. The CT interaction will appear in the r-CS region of 90° < θ < 180°. The t-HB type with covalency (t-HB_wc) appears in the region of 125° < θ_p < 150° (90° < θ < 115°), where (θ, θ_p) = (115°, 150°) is tentatively given as the borderline between t-HB_wc and the CT-MC nature. The borderline for the interactions between CT-MC and CT-TBP types is defined by θ_p = 180°. θ = 150° is tentatively given for θ_p = 180°. Classical chemical bonds of SS (180° < θ) will be strong (Cov-s) when R > 0.15 au, whereas they will be weak (Cov-w) for R < 0.15 au. The classification and characterization of interactions are summarized in Table S1 and Scheme S3 in Supplementary File.

The ^ABr-∗-^ABr and ^ABr-∗-^BBr interactions of Br₄σ(4c–6e) will be classified and characterized based on the (R, θ, θ_p) values, employing the standard values as a reference (see Scheme S2 in Supplementary File). R < 0.15 au for all interactions in Table 1; therefore, no Cov-s were detected in this work. The (θ, θ_p) values are (180.1°, 191.8°) for the original Br₂ (D_∞h) if evaluated with MP2/BSS-A. Therefore, the nature of Br-∗-Br in Br₂ (D_∞h) is classified by the SS interactions and characterized as the Cov-w nature, which is denoted by SS/Cov-w. The (θ, θ_p) values are (170.6–179.0°, 190.6–191.7°) for ^ABr-∗-^ABr of Br₄σ(4c–6e) in the optimized structures in Table 1, of which nature is r-CS/CT-TBP. The (θ, θ_p) values are (78.0–84.1°, 94.7–105.1°) for ^ABr-∗-^BBr in the optimized structures of Br₆ (C₂), Br₆ (C_2h), and R₂Br₄ (C_2h) (R = H and Me); therefore, the nature is predicted to be r-CS/t-HB_wc. The nature of ^ABr-∗-^BBr in R₄Se₂Br₆ (C_i) (R = H and Me), 5 (C_i) and 6 (C_i), is r-CS/t-HB_wc, judging from the (θ, θ_p) values of (90.9–92.8°, 116.4–122.5°). The calculated (θ, θ_p) values of ^ABr-∗-^ABr and ^ABr-∗-^BBr for the optimized structure of (D_∞h) are (170.6°, 190.6°) and (89.5°, 118.2°), respectively. In this case, ^ABr-∗-^ABr and ^ABr-∗-^BBr are predicted to have the nature of r-CS/CT-TBP and p-CS/t-HB_nc, respectively. However, ^ABr-∗-^BBr is just the borderline region to the r-CS interactions with θ = 89.5°. The characteristic nature of the ^BE---^AE-^AE---^BE interactions in (D_∞h) would be controlled by the double negative charges in the species.

The results in Table 1 show that the ^ABr-∗-^ABr interaction in Br₄σ(4c–6e) becomes weaker, as the strength of the corresponding ^ABr-∗-^BBr increases. The strength of ^ABr-∗-^ABr becomes weaker in the order shown in equation (10), if evaluated by θ, while that of ^ABr-∗-^BBr increases in the order shown in equation (11), if measured by θ. Very similar results were obtained by θ_p:

The orders shown in equations (10) and (11) seem to reasonably explain the characteristic behavior of Br₄σ(4c–6e). The results must be the reflection of the n_p(^BBr) ⟶ σ∗(^ABr-^ABr) ← n_p(^BBr) form of Br₄σ(4c–6e), where ^ABr-∗-^ABr and ^ABr-∗-^BBr become weaker and stronger, respectively, as the CT interaction increases. Br₄σ(4c–6e) will be stabilized more effectively, if the negative charge is developed more at ^BBr. However, the two Br⁻ ligands in (D_∞h) seem not so effective than that expected. This would come from the electrostatic repulsive factor between the double negative charges in (D_∞h), as mentioned above.

The θ values for (^ABr-∗-^ABr and ^ABr-∗-^BBr) in Br₆ (C_2h)_obsd and 1 (C_i)_obsd are (165.2°, 82.5°) and (175.3°, 87.7°), respectively. Therefore, ^ABr-∗-^ABr and ^ABr-∗-^BBr are classified by r-CS and p-CS, respectively. Both ^ABr-∗-^ABr and ^ABr-∗-^BBr in Br₆ (C_2h)_obsd are predicted to be weaker than those in 1 (C_i)_obsd, respectively. The results would be curious at the first glance, since ^ABr-∗-^ABr will be weaker, if ^ABr-∗-^BBr in ^BBr-∗-^ABr-∗-^ABr-∗-^BBr becomes stronger, as mentioned above. They would be affected from the surrounding, such as the crystal packing effect. A Br₂ molecule interacts with four bromine atoms adjacent to the Br₂ molecule on the bc-plane in crystals, equivalently with 3.251 Å [40].

Similar investigations were carried out for I₄σ(4c–6e), which will be discussed elsewhere (it is demonstrated that Br₄σ(4c–6e) is predicted to be somewhat stronger than I₄σ(4c–6e)).

3.6. Nature of Se₂Br₅σ(7c–10e)

The nature of Se₂Br₅σ(7c–10e) in 7 (C_2h) and 8 (C_2h) is elucidated, together with SeBr₂σ(3c–4e) in 3 and SeBr₄σ(4c–6e) in 4. The results are collected in Table 2. Figure 6 shows symmetric ψ₁₈₄ (HOMO) and antisymmetric ψ₁₈₅ (LUMO) of 8 (C_2h), which correspond to ψ₅ and ψ₆ in σ(7c–10e), illustrated in Figure 1 although the Se atoms are contained in the linear Se₂Br₅σ(7c–10e) in 8 (C_2h). The linear seven atomic orbitals on Se₂Br₅ are shown to construct ψ₁₈₄ (HOMO) and ψ₁₈₅ (LUMO) of 8 (C_2h), which can be analysed as the Se₂Br₅σ(7c–10e) [39], so can the linear interaction in 7 (C_2h), although not shown. The pseudolinear interaction of the seven atoms of 1 (C₁)_obsd could also be explained by the Se₂Br₅σ(7c–10e) model.

The results demonstrate that Se₂Br₅σ(7c–10e) stabilize well 7 (C_2h) and 8 (C_2h) although 1 (C₁)_obsd seems not so effective. The negative charge developed at the Br atom in 3 would not be sufficient to stabilize Se₂Br₅σ(7c–10e) in 1 (C₁)_obsd, relative to the case of the Br⁻ anion in 7 (C_2h) and 8 (C_2h), irrespective of the highly negatively charged Br atoms in SeBr₂σ(3c–4e) of 3.

4. Conclusion

The intrinsic dynamic and static nature of Br₄σ(4c–6e) is elucidated for 1 (C_i)_obsd and the related species with QTAIM-DFA, employing the perturbed structures generated with CIV. The ^ABr-^ABr interactions in ^BBr-∗-^ABr-∗-^ABr-∗-^BBr of Br₄σ(4c–6e) are weaker than Br-∗-Br in the optimized structure of Br₂ (D_∞h), which is predicted to have the SS/Cov-w nature. The ^ABr-^ABr interactions in Br₄σ(4c–6e) of the models are predicted to have the r-CS/CT-TBP nature, if optimized with MP2/BSS-A. The ^ABr-^ABr interaction in 1 (C_i)_obsd also appears in the r-CS region. On the contrary, the ^ABr-^BBr interactions in Br₆ (C₂), Br₆ (C_2h), H₂Br₄ (C_2h), and Me₂Br₄ (C_2h) are predicted to have the p-CS/t-HB_nc nature, whereas those in H₄Se₂Br₄ (C_i), Me₄Se₂Br₄ (C_i), 5 (C_i), and 6 (C_i) have the r-CS/t-HB_wc nature, if evaluated with MP2/BSS-A. The ^ABr-∗-^BBr interactions become stronger in the order of H₂Br₄ (C_2h) < Br₆ (C_2h) ≤ Br₆ (C₂) < Me₂Br₄ (C_2h) << Me₄Se₂Br₆ (C_i) ≤ H₄Se₂Br₆ (C_i) ≤ 5 (C_i) < 6 (C_i), which is the inverse order for ^ABr-∗-^ABr, as a whole. The results are in accordance with the CT interaction of the n_p(^BBr) ⟶ σ∗(^ABr-^ABr) ← n_p(^BBr) form derived from Br₄σ(4c–6e). The decreased binding force of ^ABr-∗-^ABr must be transferred to ^ABr-∗-^BBr in Br₄σ(4c–6e). Namely, it is demonstrated that Br₄σ(4c–6e) is stabilized as the strength of ^ABr-∗-^BBr in Br₄σ(4c–6e) increases, while ^ABr-∗-^ABr becomes weakened relative to that in the original Br₂ (D_∞h). In this process, Br₄σ(4c–6e) is totally stabilized. The ^ABr-∗-^ABr and ^ABr-∗-^BBr interactions in Br₆ (C_2h)_obsd and 1 (C_i)_obsd are classified by the r-CS and p-CS interactions, respectively, where the interactions in Br₆ (C_2h)_obsd seem somewhat weaker than those in 1 (C_i)_obsd. The Se₂Br₅σ(7c–10e) interactions are similarly elucidated for 2 (C₁)_obsd and the anionic models of 7 (C_2h) and 8 (C_2h). The Se₂Br₅σ(7c–10e) nature is clearly established for the optimized structures of 7 (C_2h) and 8 (C_2h), rather than 2 (C₁)_obsd. Extended hypervalent interactions of the σ(mc–ne: 4 ≤ m; m < n < 2m) type are shown to be well analysed and evaluated with QTAIM-DFA, employing the perturbed structures generated with CIV, exemplified by Br₄σ(4c–6e) and Se₂Br₅σ(7c–10e).

Data Availability

The data used to support the findings of this study are available in the supplementary information files.

Conflicts of Interest

The authors declare that there are no conflicts of interest regarding the publication of this paper.

Acknowledgments

This work was partially supported by a Grant-in-Aid for Scientific Research (no. 17K05785) from the Ministry of Education, Culture, Sports, Science, and Technology of Japan.

Supplementary Materials

Scheme S1: classification of interactions by the signs of ∇²ρ_b(r_c) and H_b(r_c), together with G_b(r_c) and V_b(r_c). Scheme S2: QTAIM-DFA: a plot of H_b(r_c) versus H_b(r_c) − V_b(r_c)/2 for weak to strong interactions. Scheme S3: rough classification and characterization of interactions by θ and θ_p, together with k_b(r_c) (= V_b(r_c)/G_b(r_c)). QTAIM-DFA approach, computational data (Tables S2–S5 and Figures S3–S5), computation information and geometries of compounds, and graphical abstract. Figure S1: polar (R, θ) coordinate representation of H_b(r_c) versus H_b(r_c) − V_b(r_c)/2, with (θ_p, κ_p) parameters. Figure S2: plot of H_b(r_c) versus in r(¹Cl-²Cl) = r_o(¹Cl-²Cl) + for ¹Cl-²Cl-³Cl⁻ (a) with the magnified picture of (a) (b) and that of H_b(r_c) − V_b(r_c)/2 versus (c). Typical hydrogen bonds without covalency and typical hydrogen bonds with covalency are abbreviated as t-HB without cov. and t-HB with cov., respectively, whereas Cov-w and Cov-s stand for weak covalent bonds and strong covalent bonds, respectively. Table S1: proposed definitions for the classification and characterization of interactions. (Supplementary Materials)

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Copyright © 2020 Satoko Hayashi et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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Bioinorganic Chemistry and Applications

Dynamic and Static Nature of Br4σ(4c–6e) and Se2Br5σ(7c–10e) in the Selenanthrene System and Related Species Elucidated by QTAIM Dual Functional Analysis with QC Calculations

Abstract

1. Introduction

2. Methodological Details in Calculations

3. Results and Discussion

3.1. Structural Optimizations

3.2. Energies for Formation of Br4σ(4c–6e) and NBO Analysis

3.3. Molecular Graphs with Contour Plots for the Species Containing Br4σ(4c–6e), Se2Br5σ(7c–10e), and Related Linear Interactions

3.4. Survey of Br4σ(4c–6e) and Se2Br5σ(7c–10e)

3.5. Nature of Br4σ(4c–6e)

3.6. Nature of Se2Br5σ(7c–10e)

4. Conclusion

Data Availability

Conflicts of Interest

Acknowledgments

Supplementary Materials

References

Copyright

Dynamic and Static Nature of Br₄σ(4c–6e) and Se₂Br₅σ(7c–10e) in the Selenanthrene System and Related Species Elucidated by QTAIM Dual Functional Analysis with QC Calculations

3.2. Energies for Formation of Br₄σ(4c–6e) and NBO Analysis

3.3. Molecular Graphs with Contour Plots for the Species Containing Br₄σ(4c–6e), Se₂Br₅σ(7c–10e), and Related Linear Interactions

3.4. Survey of Br₄σ(4c–6e) and Se₂Br₅σ(7c–10e)

3.5. Nature of Br₄σ(4c–6e)

3.6. Nature of Se₂Br₅σ(7c–10e)